{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2011:VHOBHSSH7POGYDVPCU7AC32U2U","short_pith_number":"pith:VHOBHSSH","canonical_record":{"source":{"id":"1107.1885","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2011-07-10T19:24:18Z","cross_cats_sorted":["math.AP"],"title_canon_sha256":"c1282a5b245d9acbd3354acafc0befc8d26b0315240a1d4bb8e1278f88e5e3c3","abstract_canon_sha256":"6bfc4bceec74a97e52bf725603aca4bfcce70fc60ed364f3f2f97ac845999e4e"},"schema_version":"1.0"},"canonical_sha256":"a9dc13ca47fbdc6c0eaf153e016f54d50ee35624e6bec4784cbf659bcd33dd2e","source":{"kind":"arxiv","id":"1107.1885","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1107.1885","created_at":"2026-05-18T04:18:33Z"},{"alias_kind":"arxiv_version","alias_value":"1107.1885v1","created_at":"2026-05-18T04:18:33Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1107.1885","created_at":"2026-05-18T04:18:33Z"},{"alias_kind":"pith_short_12","alias_value":"VHOBHSSH7POG","created_at":"2026-05-18T12:26:44Z"},{"alias_kind":"pith_short_16","alias_value":"VHOBHSSH7POGYDVP","created_at":"2026-05-18T12:26:44Z"},{"alias_kind":"pith_short_8","alias_value":"VHOBHSSH","created_at":"2026-05-18T12:26:44Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2011:VHOBHSSH7POGYDVPCU7AC32U2U","target":"record","payload":{"canonical_record":{"source":{"id":"1107.1885","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2011-07-10T19:24:18Z","cross_cats_sorted":["math.AP"],"title_canon_sha256":"c1282a5b245d9acbd3354acafc0befc8d26b0315240a1d4bb8e1278f88e5e3c3","abstract_canon_sha256":"6bfc4bceec74a97e52bf725603aca4bfcce70fc60ed364f3f2f97ac845999e4e"},"schema_version":"1.0"},"canonical_sha256":"a9dc13ca47fbdc6c0eaf153e016f54d50ee35624e6bec4784cbf659bcd33dd2e","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:18:33.952216Z","signature_b64":"J5LdjApUKICYUAFYAUhHGyzpn69VTyNoJ/LyuO67fm2E0hABpDynmDR107V0OeZ1VGgVzYcE5I95h3NLljxXBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a9dc13ca47fbdc6c0eaf153e016f54d50ee35624e6bec4784cbf659bcd33dd2e","last_reissued_at":"2026-05-18T04:18:33.951612Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:18:33.951612Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1107.1885","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T04:18:33Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"5he8WbNE08D+1sJt8WDuesbTrA4+uOiLd/Z8e9O33Q6p507EpdzCBOFg2p0ex0PTdv2IoZ+JhMhadEEebpVgDw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-27T04:22:16.739090Z"},"content_sha256":"61146101e6ddda91883b29db99f4b86212163ee0aa7991954bc82b616261abc9","schema_version":"1.0","event_id":"sha256:61146101e6ddda91883b29db99f4b86212163ee0aa7991954bc82b616261abc9"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2011:VHOBHSSH7POGYDVPCU7AC32U2U","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Sharp estimates involving $A_\\infty$ and $LlogL$ constants, and their applications to PDE","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AP"],"primary_cat":"math.CA","authors_text":"Alexander Reznikov, Oleksandra Beznosova","submitted_at":"2011-07-10T19:24:18Z","abstract_excerpt":"It is a well known fact that the union of the Reverse H\\\"{o}lder classes coincides with the union of the Muckenhoupt classes $A_p$, but the $A_\\infty$ constant of the weight $w$, which is a limit of its $A_p$ constants, is not a natural characterization for the weight in Reverse H\\\"{o}lder classes. We introduce the $RH_1$ condition as a limiting case of the $RH_p$ inequalities as $p$ tends to 1. Then we show sharp bound on $RH_1$ constant of the weight $w$ in terms of its $A_\\infty$ constant (from above and from below). We also prove the sharp version of the Gehring theorem for the case $p=1$,"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1107.1885","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T04:18:33Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"qF1XRo5gctXjlNkQGc2nnep49x+gh+LU+JKAk9o5GjJIYAoTPhXef/rSv2lugnfxPcLd08YLhl4GjlRyil8uDg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-27T04:22:16.739774Z"},"content_sha256":"890d77ce520f61d1d1f1d8b7f796e49690f68895c4fa45e9ecd6c80385d5128b","schema_version":"1.0","event_id":"sha256:890d77ce520f61d1d1f1d8b7f796e49690f68895c4fa45e9ecd6c80385d5128b"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/VHOBHSSH7POGYDVPCU7AC32U2U/bundle.json","state_url":"https://pith.science/pith/VHOBHSSH7POGYDVPCU7AC32U2U/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/VHOBHSSH7POGYDVPCU7AC32U2U/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-05-27T04:22:16Z","links":{"resolver":"https://pith.science/pith/VHOBHSSH7POGYDVPCU7AC32U2U","bundle":"https://pith.science/pith/VHOBHSSH7POGYDVPCU7AC32U2U/bundle.json","state":"https://pith.science/pith/VHOBHSSH7POGYDVPCU7AC32U2U/state.json","well_known_bundle":"https://pith.science/.well-known/pith/VHOBHSSH7POGYDVPCU7AC32U2U/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2011:VHOBHSSH7POGYDVPCU7AC32U2U","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"6bfc4bceec74a97e52bf725603aca4bfcce70fc60ed364f3f2f97ac845999e4e","cross_cats_sorted":["math.AP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2011-07-10T19:24:18Z","title_canon_sha256":"c1282a5b245d9acbd3354acafc0befc8d26b0315240a1d4bb8e1278f88e5e3c3"},"schema_version":"1.0","source":{"id":"1107.1885","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1107.1885","created_at":"2026-05-18T04:18:33Z"},{"alias_kind":"arxiv_version","alias_value":"1107.1885v1","created_at":"2026-05-18T04:18:33Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1107.1885","created_at":"2026-05-18T04:18:33Z"},{"alias_kind":"pith_short_12","alias_value":"VHOBHSSH7POG","created_at":"2026-05-18T12:26:44Z"},{"alias_kind":"pith_short_16","alias_value":"VHOBHSSH7POGYDVP","created_at":"2026-05-18T12:26:44Z"},{"alias_kind":"pith_short_8","alias_value":"VHOBHSSH","created_at":"2026-05-18T12:26:44Z"}],"graph_snapshots":[{"event_id":"sha256:890d77ce520f61d1d1f1d8b7f796e49690f68895c4fa45e9ecd6c80385d5128b","target":"graph","created_at":"2026-05-18T04:18:33Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"It is a well known fact that the union of the Reverse H\\\"{o}lder classes coincides with the union of the Muckenhoupt classes $A_p$, but the $A_\\infty$ constant of the weight $w$, which is a limit of its $A_p$ constants, is not a natural characterization for the weight in Reverse H\\\"{o}lder classes. We introduce the $RH_1$ condition as a limiting case of the $RH_p$ inequalities as $p$ tends to 1. Then we show sharp bound on $RH_1$ constant of the weight $w$ in terms of its $A_\\infty$ constant (from above and from below). We also prove the sharp version of the Gehring theorem for the case $p=1$,","authors_text":"Alexander Reznikov, Oleksandra Beznosova","cross_cats":["math.AP"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2011-07-10T19:24:18Z","title":"Sharp estimates involving $A_\\infty$ and $LlogL$ constants, and their applications to PDE"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1107.1885","kind":"arxiv","version":1},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:61146101e6ddda91883b29db99f4b86212163ee0aa7991954bc82b616261abc9","target":"record","created_at":"2026-05-18T04:18:33Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"6bfc4bceec74a97e52bf725603aca4bfcce70fc60ed364f3f2f97ac845999e4e","cross_cats_sorted":["math.AP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2011-07-10T19:24:18Z","title_canon_sha256":"c1282a5b245d9acbd3354acafc0befc8d26b0315240a1d4bb8e1278f88e5e3c3"},"schema_version":"1.0","source":{"id":"1107.1885","kind":"arxiv","version":1}},"canonical_sha256":"a9dc13ca47fbdc6c0eaf153e016f54d50ee35624e6bec4784cbf659bcd33dd2e","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"a9dc13ca47fbdc6c0eaf153e016f54d50ee35624e6bec4784cbf659bcd33dd2e","first_computed_at":"2026-05-18T04:18:33.951612Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:18:33.951612Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"J5LdjApUKICYUAFYAUhHGyzpn69VTyNoJ/LyuO67fm2E0hABpDynmDR107V0OeZ1VGgVzYcE5I95h3NLljxXBA==","signature_status":"signed_v1","signed_at":"2026-05-18T04:18:33.952216Z","signed_message":"canonical_sha256_bytes"},"source_id":"1107.1885","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:61146101e6ddda91883b29db99f4b86212163ee0aa7991954bc82b616261abc9","sha256:890d77ce520f61d1d1f1d8b7f796e49690f68895c4fa45e9ecd6c80385d5128b"],"state_sha256":"3aa1544c500987a39dacd9cf0c849b2268f554812189f3ed56acad3dac86ca85"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"xOFmqCwtDrJYq7WE1+3TSlkH1NMSIWWGcZm5iSvkCqfYN/LUM53WESufAF0DCV4jU6dOypIXiVe39RGdi2hKBQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-27T04:22:16.743560Z","bundle_sha256":"f915c64c993505a1722714a3688378c0577d1aaddcc86e7528ca0244dfd7188e"}}